Three consecutive even numbers are multiplied together. What is the largest number that is certain to divide the product, whichever three are chosen?

Write them as 2$k,2k+2$ and $2k+4$, so the product is $8\times k(k+1)(k+2)$. Any three consecutive whole numbers supply a factor of 2 and a factor of 3, hence a factor of 6, and the product always carries $8\times 6=48$. Nothing larger is safe: the very first case, $2\times 4\times 6=48$, is not a multiple of 96, which rules the biggest option out immediately. Checking a couple more, $4\times 6\times 8=192=48\times 4$ and $6\times 8\times 10=480=48\times 10$. The traps 16 and 24 do divide every such product, but the question asks for the LARGEST guaranteed divisor.